From Bombieri's Mean Value Theorem to the Riemann Hypothesis

TL;DR

This paper explores how Bombieri's mean value theorem relates to the prime number theorem and the Riemann hypothesis, establishing equivalences and bounds for primes in arithmetic progressions.

Contribution

It demonstrates the derivation of the prime number theorem and bounds on the least prime in arithmetic progressions from Bombieri's mean value theorem, linking key conjectures.

Findings

Prime number theorem is equivalent to the Riemann hypothesis under Bombieri's theorem.

Established an upper bound P(q)=O(q^2 [ln q]^32) for the least prime in arithmetic progressions.

Connected Bombieri's mean value theorem to fundamental conjectures in number theory.

Abstract

From Bombieri's mean value theorem one can deduce the prime number theorem being equivalent to the Riemann hypothesis and the least prime P(q) satisfying P(q)= O(q^2 [ln q]^32) in any arithmetic progressions with common difference q.